The chart below includes the assessed on the Algebra II California Test, the Mathematics ), the the, the Competencies in Mathematics from the Intersegmental Committee of the Academic Senate (ICAS), and the American Diploma Project (ADP) Benchmarks in Mathematics. The standards highlighted in purple are the standards assessed on the subset of CST items. The standards highlighted in green are additional standards assessed by the augmentation items. Some of the augmentation items also address standards already assessed by the CST subset (highlighted in purple). Taken together, the purple and green highlighted standards constitute the full range of measured on the. The standards not highlighted are the standards assessed on the full CST but are not selected for inclusion on the. The purpose of this crosswalk is to show if there is any alignment between the, ELM Specifications, ICAS Competencies, and ADP Benchmarks to the. It is intended to be read left-to-right. Differences in the varying levels of purpose and specificity among these sets of standards should be considered when drawing conclusions. ALGEBRA I CONTENT STANDARDS 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. A.01 Evaluate and interpret algebraic expressions A.04 Use properties of exponents N.08 Evaluate and estimate square roots Red font indicates a passage that is not addressed in the compared sets of standards. 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 4.4 Find real solutions to simple equations (including quadratic and rational), with both numerical and literal coefficients. 3.1.1. Variables, Equations, and Algebraic Expressions: J1.1. Understand the properties of Algebraic symbols and expressions; evaluation of integer exponents and roots and apply expressions and formulas; translation from words to these properties to simplify algebraic symbols; solutions of linear equations and inequalities; expressions. [ADP Core] absolute value; powers and roots; solutions of quadratic J1.2.* Understand the properties of equations; solving two linear equations in two unknowns rational exponents and apply these including the graphical interpretation of a simultaneous properties to simplify algebraic solution. Increased emphasis should be placed on expressions. algebra both as a language for describing mathematical J2.3. Understand functional notation relationships and as a means for solving problems, while and evaluate a function at a specified decreased emphasis should be placed on interpreting point in its domain. algebra as merely a set of rules for manipulating symbols. 3.3.2. Functions: Logarithmic functions, their graphs, and applications; trigonometric functions of real variables, their graphs, properties including periodicity, and applications; basic trigonometric identities; operations on functions, including addition, subtraction, multiplication, reciprocals, division, composition, and iteration; inverse functions and their graphs; domain and range. A.7. Apply laws of exponents. This CA standard generally addresses "operations" but includes a "such as" list of 5 specific skills. The ELM Topics only address 2 of them, ICAS 3, and ADP 3. Finding a reciprocal is not included in either or Specs. Missing from all four is taking an opposite. Page 1
3.0 Students solve equations and inequalities involving absolute values. A.13 Solve equations involving absolute value (in one variable) A.14 Solve inequalities involving absolute value (in one variable) 15.0 Students apply N.02 Understand and use algebraic techniques to solve percent in context rate problems, work N.05 Interpret and use ratio problems, and percent and proportion in context mixture problems. A.10 Solve problems in context that are modeled by linear equations A.16 Use constant and average rates to solve problems in context (using appropriate units) 4.6 Solve simple equations & inequalities in one variable involving absolute value. 4.4 4.4 Find real solutions to simple equations (including quadratic and rational), with both numerical and literal coefficients. Algebra II Crosswalk 3.1.1. Variables, Equations, and Algebraic Expressions: Algebraic symbols and expressions; evaluation of expressions and formulas; translation from words to symbols; solutions of linear equations and inequalities; absolute value; powers and roots; solutions of quadratic equations; solving two linear equations in two unknowns including the graphical interpretation of a simultaneous solution. Increased emphasis should be placed on algebra both as a language for describing mathematical relationships and as a means for solving problems, while decreased emphasis should be placed on interpreting algebra as merely a set of rules for manipulating symbols. J3.1. Solve linear equations and inequalities in one variable including those involving the absolute value of a linear function. [ADP Core] 3.1.1. Variables, Equations, and Algebraic Expressions: I1.2. Calculate and apply ratios, Algebraic symbols and expressions; evaluation of proportions, rates and percentages to expressions and formulas; translation from words to solve problems. symbols; solutions of linear equations and inequalities; J5.1. Recognize and solve problems absolute value; powers and roots; solutions of quadratic that can be modeled using a linear equations; solving two linear equations in two unknowns equation in one variable, such as including the graphical interpretation of a simultaneous time/rate/distance problems, solution. Increased emphasis should be placed on percentage increase or decrease algebra both as a language for describing mathematical problems, and ratio and proportion relationships and as a means for solving problems, while problems. [ADP Core] decreased emphasis should be placed on interpreting algebra as merely a set of rules for manipulating symbols. A.1. Perform arithmetic with signed numbers, including fractions and percentages. ELM Specs do not include applications with any other than quadratic functions (See 4.5). ICAS does not specifically mention ratiobased problem solving in its statements. However the examples for 3.1.1 are proportion problems. In addition examples following3.1.5 (in Data) include percentage and mixture. Page 2
22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. GEOMETRY CONTENT STANDARDS 8.0 Students know, derive, and solve problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. G.11 Graph quadratic functions in one variable G.01 Find the perimeter, area, or volume of geometric figures (including triangles, quadrilaterals, rectangular parallelepipeds, circles, cylinders, and combinations of these figures) 4.4 Find real solutions to simple equations (including quadratic and rational), with both numerical and literal coefficients. 4.2 Divide and factor polynomials (with emphasis on quadratic polynomials). 7.4 Relate basic information about a function to features of its graphs (e.g. linearity, positivity or negativity, increasing or decreasing). 5.1 Find the perimeter, area, or volume of simple geometric figures (including triangles, quadrilaterals, rectangular parallelepipeds, circles, and cylinders). 5.3 Use perimeters, areas, and volumes of simple geometric figures to find perimeters, areas and volumes of compound geometric figures. 6.2 Solve geometric problems using the properties of basic geometric figures (including triangles, quadrilaterals, polygons, and circles). 3.1.3. Geometric Concepts: Distances, areas, and volumes, and their relationship with dimension; angle measurement; similarity; congruence; lines, triangles, circles, and their properties; symmetry; Pythagorean Theorem; coordinate geometry in the plane, including distance between points, midpoint, equation of a circle; introduction to coordinate geometry in three dimensions; right angle trigonometry. Increased emphasis should be placed on developing an understanding of geometric concepts sufficient to solve unfamiliar problems and an understanding of the need for compelling geometric arguments, while decreased emphasis should be placed on memorization of terminology and formulas. A.10. Find areas of a right triangles. J4.5. Graph a quadratic function and Only ADP and ELM require understand the relationship between its graphing quadratic functions. real zeros and the x-intercepts of its Only ADP requires analysis of graph. [ADP Core] the graph and the connection J4.8. Read information and draw between the solutions and the conclusions from graphs; identify zeros. Neither or properties of a graph that provide useful Specs specifically address information about the original problem. the number of x-intercepts but [ADP Core] ELM Specs include the general statement regarding the features of a graph. K8.2. Determine the perimeter of a polygon and the circumference of a circle; the area of a rectangle, a circle, a triangle and a polygon with more than four sides by decomposing it into triangles; the surface area of a prism, a pyramid, a cone and a sphere; and the volume of a rectangular box, a prism, a pyramid, a cone and a sphere. [ADP Core] Decomposition and combinations of figures is not specifically addressed in this CA standard but is in ELM and ADP. Both and Specs include a reference to compound or complex figures not in the CA standard. Page 3
10.0* Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids. 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. G.01 Find the perimeter, area, or volume of geometric figures (including triangles, quadrilaterals, rectangular parallelepipeds, circles, cylinders, and combinations of these figures) G.02 Calculate the ratio of corresponding geometric measurements of similar figures (e.g., if the perimeters are in a 3:2 ratio, the areas are in a 9:4 ratio) 5.1 Find the perimeter, area, or volume of simple geometric figures (including triangles, quadrilaterals, rectangular parallelepipeds, circles, and cylinders). 5.2 Calculate the ratio of corresponding geometric measurements of similar figures (e.g. if the perimeters are in a 3:2 ratio, the areas are in a 9:4 ratio). 6.1 Recognize and use the properties of congruent or similar geometric objects. Algebra II Crosswalk 3.1.3. Geometric Concepts: Distances, areas, and volumes, and their relationship with dimension; angle measurement; similarity; congruence; lines, triangles, circles, and their properties; symmetry; Pythagorean Theorem; coordinate geometry in the plane, including distance between points, midpoint, equation of a circle; introduction to coordinate geometry in three dimensions; right angle trigonometry. Increased emphasis should be placed on developing an understanding of geometric concepts sufficient to solve unfamiliar problems and an understanding of the need for compelling geometric arguments, while decreased emphasis should be placed on memorization of terminology and formulas. A.10. Find areas of a right triangles. 3.1.3. Geometric Concepts: Distances, areas, and volumes, and their relationship with dimension; angle measurement; similarity; congruence; lines, triangles, circles, and their properties; symmetry; Pythagorean Theorem; coordinate geometry in the plane, including distance between points, midpoint, equation of a circle; introduction to coordinate geometry in three dimensions; right angle trigonometry. Increased emphasis should be placed on developing an understanding of geometric concepts sufficient to solve unfamiliar problems and an understanding of the need for compelling geometric arguments, while decreased emphasis should be placed on memorization of terminology and formulas. K8.2. Determine the perimeter of a polygon and the circumference of a circle; the area of a rectangle, a circle, a triangle and a polygon with more than four sides by decomposing it into triangles; the surface area of a prism, a pyramid, a cone and a sphere; and the volume of a rectangular box, a prism, a pyramid, a cone and a sphere. [ADP Core] K7. Know about the similarity of figures and use the scale factor to solve problems. K8.3. Know that the effect of a scale factor k on length, area and volume is to multiply each by k, k² and k³, respectively. This ICAS statement does not include polygons other than triangles. This ICAS statement does not address the relationship between the scale factor and the change in area and volume. Page 4
17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. G.13 Find the length or midpoint of a line segment in the coordinate plane G.08 Identify and plot points in the coordinate plane 7.5 Find the length or midpoint of a line segment in the coordinate plane. Algebra II Crosswalk 3.1.3. Geometric Concepts: Distances, areas, and volumes, and their relationship with dimension; angle measurement; similarity; congruence; lines, triangles, circles, and their properties; symmetry; Pythagorean Theorem; coordinate geometry in the plane, including distance between points, midpoint, equation of a circle; introduction to coordinate geometry in three dimensions; right angle trigonometry. Increased emphasis should be placed on developing an understanding of geometric concepts sufficient to solve unfamiliar problems and an understanding of the need for compelling geometric arguments, while decreased emphasis should be placed on memorization of terminology and formulas. 3.3.3. Geometric Concepts: Two- and three-dimensional coordinate geometry; locus problems; polar coordinates; vectors; parametric representations of curves. 3.3.4. Argumentation and Proof: Mathematical induction and formal proof. Attention should be paid to the distinction between plausible, informal reasoning and complete, rigorous demonstration. K10.2. Describe a line by a linear equation. K10.3. Find the distance between two points using their coordinates and the Pythagorean theorem. K10.4.* Find an equation of a circle given its center and radius and, given an equation of a circle, find its center and radius. K1.2. State and prove key basic theorems in geometry such as the Pythagorean theorem, the sum of the angles of a triangle is 180 degrees, and the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. [ADP Core] ADP requires proof and use but does not specifically address proofs using coordinate geometry. Proof is not required in either ELM Topics or Specs. Page 5
22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. 3.1.2. Families of Functions and Their Graphs: Applications; linear functions; quadratic and power functions; exponential functions; roots; operations on functions and the corresponding effects on their graphs; interpretation of graphs; function notation; functions in context, as models for data. Increased emphasis should be placed on various representations of functions-using graphs, tables, variables, words -and on the interplay among the graphical and other representations, while decreased emphasis should be placed on repeated manipulations of algebraic expressions. 3.1.3. Geometric Concepts: Distances, areas, and volumes, and their relationship with dimension; angle measurement; similarity; congruence; lines, triangles, circles, and their properties; symmetry; Pythagorean Theorem; coordinate geometry in the plane, including distance between points, midpoint, equation of a circle; introduction to coordinate geometry in three dimensions; right angle trigonometry. Increased emphasis should be placed on developing an understanding of geometric concepts sufficient to solve unfamiliar problems and an understanding of the need for compelling geometric arguments, while decreased emphasis should be placed on memorization of terminology and formulas. 3.2.3. Geometry: Transformational geometry, including rotations, reflections, translations, and dilations; tessellations; solid geometry; three-dimensional coordinate geometry, including lines and planes. K6. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric figures are congruent and to create and analyze geometric designs. Only ICAS and ADP include rigid motions. ADP does not specify that the coordinate plane should be used. ICAS requires use of threedimensional coordinates. Page 6
ALGEBRA II CONTENT STANDARDS 1.0* Students solve equations and inequalities involving absolute value. A.13 Solve equations involving absolute value (in one variable) A.14 Solve inequalities involving absolute value (in one variable) 2.0* Students solve systems A.08 Solve systems of linear of linear equations and equations in two unknowns inequalities (in two or three variables) by substitution, with graphs, or with matrices. 4.6 Solve simple equations and inequalities in one variable involving absolute value. 4.7 Solve simple systems of linear equations in two unknowns 7.3 Graph linear and simple quadratic functions in one variable. 3.1.1. Variables, Equations, and Algebraic Expressions: J3.1. Solve linear equations and Algebraic symbols and expressions; evaluation of inequalities in one variable including expressions and formulas; translation from words to those involving the absolute value of a symbols; solutions of linear equations and inequalities; linear function. [ADP Core] absolute value; powers and roots; solutions of quadratic equations; solving two linear equations in two unknowns including the graphical interpretation of a simultaneous solution. Increased emphasis should be placed on algebra both as a language for describing mathematical relationships and as a means for solving problems, while decreased emphasis should be placed on interpreting algebra as merely a set of rules for manipulating symbols. 3.1.1. Variables, Equations, and Algebraic Expressions: Algebraic symbols and expressions; evaluation of expressions and formulas; translation from words to symbols; solutions of linear equations and inequalities; absolute value; powers and roots; solutions of quadratic equations; solving two linear equations in two unknowns including the graphical interpretation of a simultaneous solution. Increased emphasis should be placed on algebra both as a language for describing mathematical relationships and as a means for solving problems, while decreased emphasis should be placed on interpreting algebra as merely a set of rules for manipulating symbols. 3.3.1. Variables, Equations, and Algebraic Expressions: Solutions to systems of equations, and their geometrical interpretation; solutions to quadratic equations, both algebraic and graphical; the correspondence between roots and factors of polynomials; the binomial theorem. J3.3. Solve systems of two linear equations in two variables. [ADP Core] J3.4.* Solve systems of three linear equations in three variables. J5.2. Recognize and solve problems that can be modeled using a system of two equations in two variables, such as mixture problems. J3.3. Solve systems of two linear equations in two variables. [ADP Core] J3.4.* Solve systems of three linear equations in three variables. J5.2. Recognize and solve problems that can be modeled using a system of two equations in two variables, such as mixture problems. The ELM Specs include the modifier "simple," which might be seen as a limitation that does not exist in the CA standard or in the ELM Topics. ICAS and ELM do not specify equations and inequalities in three variables. In addition ELM does not address systems of inequalities. And again the word "simple" is included in the ELM Spec, a limitation is not in the topics or CA standard. Matrices are not required in any of the comparative sets of standards. Page 7
3.0* Students are adept at operations on polynomials, including long division. A.05 Perform polynomial arithmetic (add, subtract, multiply, divide, and factor) 4.0* Students factor A.05 Perform polynomial polynomials representing the arithmetic (add, subtract, difference of squares, perfect multiply, divide, and factor) square trinomials, and the sum and difference of two cubes. 5.0* Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane. 6.0* Students add, subtract, multiply, and divide complex numbers. 4.2 Divide and factor polynomials (with emphasis on quadratic polynomials). 4.2 Divide and factor polynomials (with emphasis on quadratic polynomials). 3.3.2. Functions: Logarithmic functions, their graphs, and applications; trigonometric functions of real variables, their graphs, properties including periodicity, and applications; basic trigonometric identities; operations on functions, including addition, subtraction, multiplication, reciprocals, division, composition, and iteration; inverse functions and their graphs; domain and range. A.3. Use the distributive law for monomials and binomials. J1.3. Add, subtract and multiply polynomials; divide a polynomial by a low degree polynomial. J1.4. Factor polynomials by removing the greatest common factor; factor quadratic polynomials. While long division is not specifically mentioned, division is clearly required. Polynomial operations in the ELM Specs only include division. (Factoring would be a form of division, as well.) While all comparative sets of standards address factoring polynomials, ELM, ICAS, and ADP do not specifically mention the difference of squares or the sum/difference of cubes. I3. Understand that to solve certain Only ADP addresses an problems and equations, number understanding of real and systems need to be extended from complex numbers. ADP does whole numbers to the set of all integers not require plotting on the (positive, negative and zero), from complex plane. integers to rational numbers, from rational numbers to real numbers (rational and irrational numbers) and from real numbers to complex numbers; define and give examples of each of these types of numbers. None of the other sets of standards addresses operations with complex numbers. Page 8
7.0* Students add, subtract, A.01 Evaluate and interpret multiply, divide, reduce, and algebraic expressions evaluate rational expressions A.02 Simplify algebraic with monomial and expressions polynomial denominators A.04 Use properties of and simplify complicated exponents rational A.05 Perform polynomial expressions, including those arithmetic (add, subtract, with negative exponents in multiply, divide, and factor) the denominator. A.06 Perform arithmetic operations involving rational expressions 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 4.3 Perform simple arithmetic operations involving rational expressions (add, subtract, multiply, and divide). 4.2 Divide and factor polynomials (with emphasis on quadratic polynomials). 3.1.1. Variables, Equations, and Algebraic Expressions: Algebraic symbols and expressions; evaluation of expressions and formulas; translation from words to symbols; solutions of linear equations and inequalities; absolute value; powers and roots; solutions of quadratic equations; solving two linear equations in two unknowns including the graphical interpretation of a simultaneous solution. Increased emphasis should be placed on algebra both as a language for describing mathematical relationships and as a means for solving problems, while decreased emphasis should be placed on interpreting algebra as merely a set of rules for manipulating symbols. A.2. Combine like terms in algebraic expressions. A.3. Use the distributive law for monomials and binomials. A.4. Factor monomials out of algebraic expressions. A.7. Apply laws of exponents. J1.3. Add, subtract and multiply polynomials; divide a polynomial by a low degree polynomial. J1.5. Add, subtract, multiply, divide and simplify rational expressions. J1.6. Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified values of their variables. J2.4.* Combine functions by composition, as well as by addition, subtraction, multiplication and division. J1.1. Understand the properties of integer exponents and roots and apply these properties to simplify algebraic expressions. [ADP Core] ICAS does not clearly call for operations with rational numbers. However, the requirements in the appendix imply those operations with like terms using addition and subtraction; distribution implies multiplication; and factoring implies division. Negative exponents are also not specifically mentioned in ICAS. Page 9
8.0* Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. A.11 Solve quadratic and rational equations (with both numerical and literal coefficients; real solutions only) A.12 Solve problems in context that are modeled by quadratic equations G.11 Graph quadratic functions in one variable 4.4 Find real solutions to simple equations (including quadratic and rational), with both numerical and literal coefficients. 4.5 Solve problems in context that are modeled by quadratic equations. 5.1 Find the perimeter, area, or volume of simple geometric figures (including triangles, quadrilaterals, rectangular parallelepipeds, circles, and cylinders). 6.2 Solve geometric problems using the properties of basic geometric figures (including triangles, quadrilaterals, polygons, and circles. 7.3 Graph linear and simple quadratic functions in one variable. Algebra II Crosswalk 3.3.1. Variables, Equations, and Algebraic Expressions: Solutions to systems of equations, and their geometrical interpretation; solutions to quadratic equations, both algebraic and graphical; the correspondence between roots and factors of polynomials; the binomial theorem. 3.1.2. Families of Functions and Their Graphs: Applications; linear functions; quadratic and power functions; exponential functions; roots; operations on functions and the corresponding effects on their graphs; interpretation of graphs; function notation; functions in context, as models for data. Increased emphasis should be placed on various representations of functions-using graphs, tables, variables, words -and on the interplay among the graphical and other representations, while decreased emphasis should be placed on repeated manipulations of algebraic expressions. J3.5. Solve quadratic equations in one variable. [ADP Core] J4.5. Graph a quadratic function and understand the relationship between its real zeros and the x-intercepts of its graph. [ADP Core] J5.3. Recognize and solve problems that can be modeled using a quadratic equation, such as the motion of an object under the force of gravity. [ADP Core] Solving rational equations is not required in this CA standard but is in ELM. Specific methods for finding solutions of quadratic equations is not included in other standards but is in this CA standard. Page 10
9.0* Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) 2 + c. 10.0* Students graph quadratic functions and determine the maxima, minima, and zeros of the function. 7.4 Relate basic information about a function to features of its graphs (e.g. linearity, positivity or negativity, increasing or decreasing). 7.3 Graph linear and simple quadratic functions in one variable. 7.4 Relate basic information about a function to features of its graphs (e.g. linearity, positivity or negativity, increasing or decreasing). J4.5. Graph a quadratic function and understand the relationship between its real zeros and the x-intercepts of its graph. [ADP Core] J4.5. Graph a quadratic function and understand the relationship between its real zeros and the x-intercepts of its graph. [ADP Core] J4.8. Read information and draw conclusions from graphs; identify properties of a graph that provide useful information about the original problem. [ADP Core] ADP is the only one that addresses the connection between a quadratic graph and the equation. ADP does not specifically require that students know how the graph of the parabola changes as coefficients are changed. ELM Specs includes a requirement for basic information about a graph of a function which may include how the function would be altered when parameters are changed. and ICAS do not address interpretation of a quadratic graph. This CA standard has no counterpart in but does in ELM Specs. Page 11
11.0 Students prove simple laws of logarithms. 3.3.2. Functions: Logarithmic functions, their graphs, and applications; trigonometric functions of real variables, their graphs, properties including periodicity, and applications; basic trigonometric identities; operations on functions, including addition, subtraction, multiplication, reciprocals, division, composition, and iteration; inverse functions and their graphs; domain and range. 3.3.4. Argumentation and Proof: Mathematical induction and formal proof. Attention should be paid to the distinction between plausible, informal reasoning and complete, rigorous demonstration. J2.6.* Know the inverse of an exponential function is a logarithm, prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems. MR1. Using inductive and deductive reasoning to arrive at valid conclusions. MR3. Understanding the role of definitions, proofs and counterexamples in mathematical reasoning; constructing simple proofs. [ADP Core] While ICAS requires proof in general and knowledge of logarithmic functions, there is no clear call for proof of log properties in ICAS. 11.1* Students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. A.04 Use properties of exponents 3.3.2. Functions: Logarithmic functions, their graphs, and applications; trigonometric functions of real variables, their graphs, properties including periodicity, and applications; basic trigonometric identities; operations on functions, including addition, subtraction, multiplication, reciprocals, division, composition, and iteration; inverse functions and their graphs; domain and range. A.7. Apply laws of exponents. J2.6.* Know the inverse of an exponential function is a logarithm, prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems. J5.5.* Recognize and solve problems that can be modeled using an exponential function but whose solution requires facility with logarithms, such as exponential growth and decay problems. and Specs require use of exponent properties but does not specify that log properties are to be used at all. There are no references to the inverse relationship between exponential and logarithmic functions in ELM Specs. ICAS does not make the connection between exponents and logs. Page 12
11.2* Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step. NOT INCLUDED IN THE REVIEWED BLUEPRINT. INCLUDED ON IF ITEM IS ABOUT REAL NUMBERS OR EXPONENTS. NOT INCLUDED IF ITEM IS ABOUT PROOFS OR LOGARITHMS. 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 3.1.6. Argumentation and Proof: Mathematical implication; hypotheses and conclusions; direct and indirect reasoning; inductive and deductive reasoning. Increased emphasis should be placed on constructing and recognizing valid mathematical arguments, while decreased emphasis should be placed on mathematical proofs as formal exercises. A.7. Apply laws of exponents. MR1. Using inductive and deductive reasoning to arrive at valid conclusions. J1.1. Understand the properties of integer exponents and roots and apply these properties to simplify algebraic expressions. [ADP Core] J2.6.* Know the inverse of an exponential function is a logarithm, prove basic properties of a logarithm using properties of its inverse and apply those properties to solve problems. ICAS and ADP do not specifically address the ability to judge the validity of arguments. Neither also requires the properties of real numbers to be used as the basis of judgment. The ELM Spec is used here as interpretation of the expressions that occur at each step in a logical argument. This would be another example of an ELM Spec that addresses the CA standard where there are no counterparts in. The ELM Spec is used here as interpretation of the expressions that occur at each step in a logical argument. This would be another example of an ELM Spec that addresses the CA standard where there are no counterparts in. Page 13
12.0* Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. 13.0 Students use the definition of logarithms to translate between logarithms in any base. 14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values. A.04 Use properties of exponents 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 4.4 Find real solutions to simple equations (including quadratic and rational), with both numerical and literal coefficients. 3.1.1. Variables, Equations, and Algebraic Expressions: J1.2.* Understand the properties of Algebraic symbols and expressions; evaluation of rational exponents and apply these expressions and formulas; translation from words to properties to simplify algebraic symbols; solutions of linear equations and inequalities; expressions. absolute value; powers and roots; solutions of quadratic J4.7. Graph exponential functions and equations; solving two linear equations in two unknowns identify their key characteristics. [ADP including the graphical interpretation of a simultaneous Core] solution. Increased emphasis should be placed on J5.4. Recognize and solve problems algebra both as a language for describing mathematical that can be modeled using an relationships and as a means for solving problems, while exponential function, such as decreased emphasis should be placed on interpreting compound interest problems. [ADP algebra as merely a set of rules for manipulating Core] symbols. A.7. Apply laws of exponents. While properties of exponents are required in all of the comparative sets of standards, there is no clear reference to rational exponents in or ICAS and no requirement that addresses exponential function models in either the ELM Specs or Topics. J2.6.* Know the inverse of an ADP requires knowledge and exponential function is a logarithm, application of log properties, prove basic properties of a logarithm which implies change of base. using properties of its inverse and apply those properties to solve problems. J2.6.* Know the inverse of an This CA standard is exponential function is a logarithm, specifically aimed at the prove basic properties of a logarithm properties of logarithms. Only using properties of its inverse and apply ADP addresses logarithms those properties to solve problems. specifically. The ELM Topic is very general and only refers to the simplification part of the CA standard. Page 14
15.0* Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true. 16.0 Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it. 17.0 Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation. 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 4.3 Perform simple arithmetic operations involving rational expressions (add, subtract, 4.5 Solve problems in context that are modeled by quadratic equations. 3.4.3.* Conic Sections: Representations as plane sections of a cone; focus-directory properties; reflective properties. MR1. Using inductive and deductive reasoning to arrive at valid conclusions. The ELM Specs used here reference interpretation of an expression, which may include determination of the possible degrees of truth. It is likely that the most common "algebraic statement" would be one involving operations. J4.6.* Graph ellipses and hyperbolas ICAS does not address the whose axes are parallel to the x and y relationship between the axes and demonstrate understanding of graph and the equation nor the relationship between their standard the equations of conics. algebraic form and their graphical characteristics. J4.8. Read information and draw conclusions from graphs; identify properties of a graph that provide useful information about the original problem. [ADP Core] J4.6.* Graph ellipses and hyperbolas The graph analysis whose axes are parallel to the x and y addressed in the CA standard axes and demonstrate understanding of is not clearly included in the the relationship between their standard ADP Benchmarks. algebraic form and their graphical characteristics. J4.8. Read information and draw conclusions from graphs; identify properties of a graph that provide useful information about the original problem. [ADP Core] K10.4.* Find an equation of a circle given its center and radius and, given an equation of a circle, find its center and radius. Page 15
18.0* Students use fundamental counting principles to compute combinations and permutations. 3.1.4 Probability: Counting (permutation and combinations, multiplication principle); sample spaces; expected value; conditional probability; area representation of probability. L4.4 Apply probability concepts such as conditional probability and independent events to calculate simple probabilities. The ADP Benchmark subsumes this CA standard, although it does not specifically mention counting strategies. 19.0* Students use combinations and permutations to compute probabilities. L4.4 Apply probability concepts such as conditional probability and independent events to calculate simple probabilities. ADP addresses probability calculations. 20.0* Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers. 3.3.1. Variables, Equations, and Algebraic Expressions: Solutions to systems of equations, and their geometrical interpretation; solutions to quadratic equations, both algebraic and graphical; the correspondence between roots and factors of polynomials; the binomial theorem. J6.* Understand the binomial theorem and its connections to combinatorics, Pascal s triangle and probability. ICAS and ADP include the use of the binomial theorem. ICAS does not clarify how the binomial theorem is to be applied. 21.0 Students apply the method of mathematical induction to prove general statements about the positive integers. 22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series. 23.0* Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series. 3.3.4. Argumentation and Proof: Mathematical induction and formal proof. Attention should be paid to the distinction between plausible, informal reasoning and complete, rigorous demonstration. 3.2.2. Sequences and Series: Geometric and arithmetic sequences and series; the Fibonacci sequence; recursion relations. 3.2.2. Sequences and Series: Geometric and arithmetic sequences and series; the Fibonacci sequence; recursion relations. Page 16 MR1. Using inductive and deductive reasoning to arrive at valid conclusions. J1.7.* Derive and use the formulas for the general term and summation of finite arithmetic and geometric series; find the sum of an infinite geometric series whose common ratio, r, is in the interval (-1, 1). J1.7.* Derive and use the formulas for the general term and summation of finite arithmetic and geometric series; find the sum of an infinite geometric series whose common ratio, r, is in the interval (-1, 1). ICAS includes mathematical induction. While ADP addresses inductive reasoning, it is not clear that proof by mathematical induction is implied. While ICAS addresses geometric and arithmetic sequences and series, it is unclear whether their formulas are required. While it is somewhat likely that ICAS will require using the formulas, it is less clear that deriving the formulas is required. CST Alg 2/HSM include this concept in addition to ADP.
24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions. 25.0 Students use properties from number systems to justify steps in combining and simplifying functions. NOT INCLUDED IN THE REVIEWED BLUEPRINT BUT HAS BEEN IDENTIFIED BY CSU FOR INCLUSION ON. A.02 Simplify algebraic expressions 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 4.1 Evaluate & interpret algebraic expressions (including function notation, rational exponents, and radicals). 3.3.2. Functions: Logarithmic functions, their graphs, and applications; trigonometric functions of real variables, their graphs, properties including periodicity, and applications; basic trigonometric identities; operations on functions, including addition, subtraction, multiplication, reciprocals, division, composition, and iteration; inverse functions and their graphs; domain and range. J2.4.* Combine functions by composition, as well as by addition, subtraction, multiplication and division. J2.5.* Identify whether a function has an inverse and when functions are inverses of each other; explain why the graph of a function and its inverse are reflections of one another over the line y = x. Only and Specs does not address these function operations. I3. Understand that to solve certain and Specs problems and equations, number require simplification but not systems need to be extended from justification of the steps in the whole numbers to the set of all integers process. However, (positive, negative and zero), from interpretation of an integers to rational numbers, from expression might include rational numbers to real numbers justification of the steps in (rational and irrational numbers) and simplification of a combination from real numbers to complex numbers; of functions. define and give examples of each of these types of numbers. Page 17
PROBABILITY AND STATISTICS CONTENT STANDARDS 1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces. 7.0 Students compute the variance and the standard deviation of a distribution of data. 3.1.4. Probability: Counting (permutations and combinations, multiplication principle); sample spaces; expected value; conditional probability; area representations of probability. Increased emphasis should be placed on a conceptual understanding of discrete probability, while decreased emphasis should be placed on aspects of probability that involve memorization and rote application of formulas. L4.4. Apply probability concepts such as conditional probability and independent events to calculate simple probabilities. L4.5. Apply probability concepts to practical situations to make informed decisions. [ADP Core] L4.4. Apply probability concepts such as conditional probability and independent events to calculate simple probabilities. L1.3. Compute and explain summary statistics for distributions of data including measures of center (mean, median) and spread (range, percentiles, variance, standard deviation). [ADP Core] Complementation is not addressed specifically in ADP. ELM does not address computation of conditional probabilities. Page 18